%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
\subsection{The assumption of radiative balance}
\label{Sec:RadiativeBalance}
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsubsection{Introduction}
%
Like the physical mechanism in glass houses 
the ${\rm CO}_2$-greenhouse effect is about 
a comparison of two different physical situations.
Unfortunately, the exact definition of the
atmospheric greenhouse effect changes 
from audience to audience, that is, 
there are many variations of the theme.
Nevertheless, one common aspect 
lies in the methodology that 
a fictitious model computation 
for a celestial body \textit{without} an atmosphere 
is compared to 
another fictitious model computation
for a celestial body \textit{with} an atmosphere. 
For instance, \lq\lq average\rq\rq\ temperatures
are calculated 
for an Earth \textit{without} an atmosphere 
and 
for an Earth \textit{with} an atmosphere.
Amusingly, there seem to exist no calculations 
for an Earth  \textit{without} oceans 
opposed to  calculations 
for an Earth \textit{with} oceans. 
However, in many studies, models for oceanic currents 
are included in the frameworks considered, 
and radiative \lq\lq transport\rq\rq\ 
calculations are incorporated too. 
Not all of these refinements can be 
discussed here in detail. The reader is referred to
%
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%
\ifthenelse{\equal{IJMPB}{\TschStyle}}
           {Ref.~\refcite{McGuffie2006}}
           {}%
\ifthenelse{\equal{arXiv}{\TschStyle}}
           {Ref.~\cite{McGuffie2006}}
           {} 
\ifthenelse{\equal{TeX4ht}{\TschStyle}}
           {Ref.~\cite{McGuffie2006}}
           {} 
%       
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%
and further references therein.
Though there exists a huge family of 
generalizations, one common aspect is the assumption
of a radiative balance, which plays a central role
in the publications of the IPCC 
and, hence, in the public propaganda.
In the following it is proved that this assumption
is physically wrong.
%
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%
\subsubsection{A note on \lq\lq radiation balance\rq\rq\ diagrams}
%
From the definition given in Section~\ref{Sec:ISI}  
it is immediately evident that 
a radiation intensity 
$I_\nu$
is \textit{not} a current density 
that can be described by a vector field 
$\textbf{j}(\textbf{x},t)$.
That means that conservation laws
(continuity equations, 
balance equations, 
budget equations)
cannot be written down for intensities.
Unfortunately this is done in most
climatologic papers, 
\textbf{the cardinal error of global climatology},
that may have been overlooked so long due to
the oversimplification of the 
real world problem towards a quasi 
one-dimensional problem.
Hence the popular climatologic 
\lq\lq radiation balance\rq\rq\ diagrams 
describing quasi-one-dimensional situations
(cf.\ Figure~\ref{Fig:Diagrams}) 
are scientific misconduct since they do not properly represent 
the mathematical and physical fundamentals.
%
\begin{figure}[htbp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_EarthsEnergyBalance_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=0.80]{PictureSmall_EarthsEnergyBalance_.png}}
\vspace*{-8pt}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=0.80]{PictureLarge_EarthsEnergyBalance_.png}}
\vspace*{-8pt}
}{}
\vspace*{8pt}
\caption{A schematic diagram supposed to describe 
         the global average components of the Earth's 
         energy balance.       
         Diagrams of this kind \textbf{contradict to physics.}}
%
\label{Fig:Diagrams}
\end{figure}
%

Diagrams of the type of Figure~\ref{Fig:Diagrams}
are the cornerstones of \lq\lq climatologic proofs\rq\rq\ 
of the supposed Greenhouse effect in the atmosphere\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{MacCracken1985}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
They are highly suggestive, because they bear some 
similarity to Kirchhoff rules of electrotechnics,
in particular to the node rule describing the conservation
of charge\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Paul2001}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Unfortunately, in the literature on global 
climatology it is not explained, what the
arrows  in \lq\lq radiation balance\rq\rq\ 
diagrams mean physically. It is easily 
verified that within the frame of physics 
they cannot mean anything.

Climatologic radiation balance diagrams are nonsense,
since they  
%
\begin{enumerate}
\item cannot represent radiation intensities,
      the most natural interpretation of the arrows
      depicted in Figure~\ref{Fig:Diagrams}, 
      as already explained in 
      Section~\ref{Sec:ISI} 
      and 
      Section~\ref{Sec:Conclusion} ;
\item cannot represent sourceless fluxes,  
      i.e.\ a divergence free vector fields
      in three dimensions, since a vanishing 
      three-dimensional divergence still allows
      that a portion of the field goes sidewards;  
\item do not fit in the framework of Feynman diagrams,
      which represent mathematical expressions clearly
      defined in quantum field theory\TschSpace%   
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Itzykson1980}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                   
\item do not fit in the standard language
      of system theory or system engineering\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{SysML2007}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\end{enumerate}
%
Kirchhoff-type node rules only hold in cases, 
where there is a conserved quantity and the 
underlying space may be described by a topological 
space that is a one-dimensional manifold almost 
everywhere, the singularities being the 
network nodes,
i.e. in conventional electric circuitry
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Paul2001},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
in mesoscopic networks\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Balachandran1992},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
and, 
for electromagnetic waves, 
in waveguide networks%
%
\footnote{%
The second and the third type are beautifully related 
by the correspondence of the v.\,Klitzing resistance 
${\rm R}_{\rm vK} \approx 25,813\ {\rm k}\Omega$
with the characteristic impedance
${\rm Z}_0 \approx 376,73\ \Omega$
via the Sommerfeld fine structure constant
$\alpha={\rm Z}_0/2{\rm R}_{\rm vK} \approx 1/137,036\,$\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Tscheuschner1998}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
}%
\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Montgomery1948,Marcuvitz1986}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
However, although Kirchhoff's mesh analysis 
may be successfully applied to microwave networks, 
the details are highly involved 
and will break down if dissipation is allowed\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Montgomery1948,Marcuvitz1986}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Clearly, neither the cryptoclimate of a glass house 
nor the atmosphere of the Earth's does compare 
to a waveguide network e.g.\ feeding the
acceleration cavities of a particle accelerator.
Therefore, the climatologic radiation balance  
diagrams are inappropriate and misleading,
even when they are supposed to describe 
averaged quantities. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsubsection{The case of purely radiative balance}
%
If only thermal radiation was possible for 
the heat transfer of a radiation-exposed body 
one would use Stefan-Boltzmann's law
%
\begin{equation}
{\rm S}(T) = \sigma T^4
\label{Eq:epsilonsigmaTto4}
\end{equation}
%
to calculate the ground temperature determined by this balance. The irradiance ${\rm S}$ has dimensions of 
a power density and $\sigma$ is the Stefan-Boltzmann 
constant given by 
%
\begin{equation}
\sigma 
=
\frac{2 \pi^5 k^4}{15 c^2 h^3} 
=
5.67040ß0 \cdot 10^{-8} \, 
\frac{{\rm W}}{{\rm m}^2{\rm K}^4}
\approx
5.67
\cdot
\left(
\frac{T}{100}
\right)^4
\,
\frac{{\rm W}}{{\rm m}^2{\rm K}^4}
\end{equation}
% 
For example, the energy flux density of 
a black body a room temperature $300\,{\rm K}$
is approximately
%
\begin{equation}
{\rm S}(\,T\!=\!300\,{\rm K}\,) = 459\ {\rm W}/{\rm m}^2
\end{equation}
%

One word of caution is needed here:
As already emphasized in Section~\ref{Sec:Conclusion}
the constant $\sigma$ appearing in the $T^4$ law
is \textit{not} a universal constant of physics.
Furthermore, a grey radiator 
must be described by a temperature dependent 
$\sigma(T)$ spoiling the $T^4$ law. 
Rigorously speaking, for real objects the 
Equation (\ref{Eq:epsilonsigmaTto4}) 
is invalid.
Therefore all crude approximations 
relying on $T^4$ expressions need 
to be taken with great care.
In fact, though popular in global
climatology, they prove nothing!  

In the balance equation
%
\begin{equation}
\sigma
\cdot
T_{\mbox{{\scriptsize\rm Earth's ground}}}^4 
=
\sigma
\cdot
T_{\mbox{{\scriptsize\rm Sun}}}^4
\cdot 
\frac{{\rm R}^2_{\mbox{\scriptsize\rm Sun}}}
     {{\rm R}^2_{\mbox{\scriptsize\rm Earth's orbit}}}
\end{equation}
% 
one may insert 
a general phenomenological normalization factor
$\epsilon$
at the right side, leaving room
for a fine tuning and inclusion of geometric
factors.%
\footnote{%
The factor 
$\varepsilon$
is related to 
the albedo 
$A$
of the Earth
describing 
her reflectivity:
$A = 1 - \varepsilon$.
In the earlier literature 
one often finds 
$A=0.5$ 
for the Earth,
in current publications 
$A=0.3$.
The latter value 
is used here.%
}
Thus one may write
%
\begin{equation}
\sigma \cdot T^4_{\mbox{{\scriptsize\rm Earth's ground}}}
=
\epsilon \cdot \sigma \cdot 5780^4 \cdot \frac{1}{46225}
=
\epsilon \cdot 1368\ {\rm W}/{\rm m}^2
=
\epsilon \cdot {\rm s}
\end{equation}
%
which yields
%
\begin{equation}
T_{\mbox{{\scriptsize\rm Earth's ground}}}
=
\sqrt[4]{\epsilon} \cdot \frac{\,\,5780}{\,\,\sqrt{215}}\ {\rm K} 
=
\sqrt[4]{\epsilon} \cdot 394.2\ {\rm K}
\label{Eq:TE75}
\end{equation}
%
${\rm s}$ is the solar constant.
With the aid of 
Equation~(\ref{Eq:TE75}) 
one calculates the values 
displayed in 
Table~\ref{temps}.

%
\input{GerlichTscheuschner2007a_370_co2_greenhouses_radiative_table010_.tex}
%
Only the temperature measured in the Sun inside 
the car bears some similarity 
with the three ones calculated here. 
Therefore, the radiation balance does 
not determine the temperature outside the car!
In contrast to this, Table~\ref{avertemps} 
displays the \lq\lq average effective\rq\rq\ temperatures
of the ground, which according to climatological consensus
are used to \lq\lq explain\rq\rq\ the atmospheric
greenhouse effect.
The factor of a quarter is introduced 
by \lq\lq distributing\rq\rq\ 
the incoming solar radiation 
seeing a cross section $\sigma_{\mbox{{\scriptsize\rm Earth}}}$
over the global surface $\Omega_{\mbox{{\scriptsize\rm Earth}}}$
%
\begin{equation}
\frac{ \sigma_{\mbox{{\scriptsize\rm Earth}}} }
     { \Omega_{\mbox{{\scriptsize\rm Earth}}} }
=
\frac{ \,\,\pi \cdot {\rm R}^2_{\mbox{\scriptsize\rm Earth}} }  
     {    4\pi \cdot {\rm R}^2_{\mbox{\scriptsize\rm Earth}} }
=
\frac{1}{4}    
\label{Eq:quarter}     
\end{equation}
%
\input{GerlichTscheuschner2007a_370_co2_greenhouses_radiative_table011_.tex}
%
The fictitious natural greenhouse effect 
is the difference the 
\lq\lq average effective\rq\rq\ temperature 
of 
$-18\,^\circ{\rm C}$
and the
Earth's 
\lq\lq observed\rq\rq\
average temperature
of
$+15\,^\circ{\rm C}$.

In summary, 
the factor 0.7 will enter the equations 
if one assumes that 
a grey body absorber 
is a black body radiator,
contrary to the laws of physics.
Other choices are possible,
the result is arbitrary.
Evidently, such an average value 
has no physical meaning at all.
This will be elucidated in the following 
subsection.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
\subsubsection{The average temperature of a radiation-exposed globe}
%
\label{Sec:AverageTemperature}
%
\begin{figure}[htbp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_Earthball_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_Earthball_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_Earthball_.png}}
}{}
\vspace*{8pt}
\caption{A radiation exposed static globe.}
\label{globe}
\end{figure}
%
For a radiation exposed static globe 
(cf.\ Figure~\ref{globe})
the corresponding balance equation 
must contain a geometric factor and 
reads therefore
%
\begin{equation}
\sigma \cdot T^4
=
\left\{\begin{array}{cl}
\epsilon \cdot {\rm S} \cdot \cos\, \vartheta 
=
\epsilon \cdot \sigma \cdot 5780^4 / 215^2 \cdot \cos \vartheta
&
\mbox{{\rm \ \ \ if\ \ }}
\phantom{\pi/}0\le\vartheta\leq\pi/2\phantom{}
\\
0
& 
\mbox{{\rm \ \ \ if\ \ }}
\phantom{}\pi/2\le\vartheta\leq\pi\phantom{/2}
\end{array}\right.
\end{equation}
%
It is obvious that one gets 
the effective temperatures
if the right side is divided by $\sigma$.

This in turn will determine the formerly mentioned 
\lq\lq average\rq\rq\ effective temperatures
over the global surface.
%
\begin{eqnarray}
T^4_{\mbox{\scriptsize\rm eff}} 
&=&
\frac{1}{4\pi} \int\!\!\!\int_{\mbox{\scriptsize\rm surface}} T^4
\,d\Omega
\nonumber \\
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{0}^{\pi} T^4 
\sin\vartheta\,d\vartheta\,d\varphi
\nonumber \\
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{1}^{-1} T^4 
d(-\cos\vartheta)\,d\varphi
\nonumber \\
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{-1}^{1} T^4 
d(\cos\vartheta)\,d\varphi
\end{eqnarray}
%
Defining 
%
\begin{equation}
\mu := \cos\vartheta
\end{equation}
one gets 
%
\begin{eqnarray}
T^4_{\mbox{\scriptsize\rm eff}} 
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{-1}^{1} T^4 \, d\mu \, d\varphi
\nonumber \\
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{0}^{1} \epsilon \cdot 
\frac{{\rm S}}{\sigma} \cdot \mu \, d\mu \, d\varphi
\nonumber \\
&=&
\frac{1}{2} \cdot \epsilon \cdot \frac{{\rm S}}{\sigma} \cdot \int_{0}^{1} \, \mu \, d\mu
\nonumber \\
&=&
\frac{1}{4} \cdot \epsilon \cdot \frac{S}{\sigma}
\nonumber \\
&=&
\frac{1}{4} \cdot \epsilon \cdot (394.2)^4\ {\rm K}^4
\end{eqnarray}
%
This is the correct derivation of the factor quarter
appearing in Equation (\ref{Eq:quarter}). 
Drawing the fourth root 
out of the resulting expression   
%
\begin{eqnarray}
T_{\mbox{\scriptsize\rm eff}} 
&=&
\sqrt[4]
{ 
\frac{\epsilon}{4} \cdot \frac{{\rm S}}{\sigma} 
}
\nonumber \\
&=&
\sqrt[4]
{ 
\frac{\epsilon}{4}
} 
\cdot 394.2\ {\rm K} 
\nonumber \\
&=&
(1 / \sqrt{2})
\cdot
\sqrt[4]{\epsilon}
\cdot 394.2\ {\rm K}
\nonumber \\
&=&
0.707
\cdot
\sqrt[4]{\epsilon}
\cdot 
394.2\ {\rm K}
\end{eqnarray}
%
Such a calculation, 
though standard in global climatology,
is plainly wrong. 
Namely, if one wants to calculate 
the average temperature, 
one has to draw the fourth root first and 
then determine the average, though:
%
\begin{eqnarray}
T_{\mbox{\scriptsize\rm phys}} 
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{-1}^{1} T \, d\mu \, d\varphi
\nonumber \\
&=&
\frac{1}{4\pi} \int_{0}^{2\pi} \int_{0}^{1} 
\sqrt[4]
{
\epsilon 
\cdot 
\frac{{\rm S}}{\sigma} \cdot \mu
} 
\,\,\,d\mu \, d\varphi
\nonumber \\
&=&
\frac{1}{2}
\cdot 
\sqrt[4]
{
\epsilon 
\cdot 
\frac{{\rm S}}{\sigma}
} 
\cdot
\int_{0}^{1} 
\sqrt[4]{\mu}
\,\,\,
d\mu
\nonumber \\
&=&
\frac{1}{2}
\cdot 
\sqrt[4]
{
\epsilon 
\cdot 
\frac{{\rm S}}{\sigma}
} 
\cdot
\frac{4}{5}
\nonumber \\
&=&
\frac{2}{5}
\cdot 
\sqrt[4]
{
\epsilon 
\cdot 
\frac{{\rm S}}{\sigma}
} 
\end{eqnarray}
%
\noindent%
finally yielding 
%
\begin{eqnarray}
T_{\mbox{\scriptsize\rm phys}} 
&=&
\frac{2}{5}
\cdot
\sqrt[4]{\epsilon}
\cdot
394.2\ {\rm K}
\nonumber \\
&=&
0.4
\cdot
\sqrt[4] {\epsilon}
\cdot
394.2\ {\rm K}
\label{Eq:Tphys}
\end{eqnarray}
%
Now the averaged temperatures 
$T_{\mbox{\scriptsize\rm phys}}$ 
are considerably lower 
than the absolute temperature's fourth 
root of the averaged fourth power
(cf.\ Table~\ref{phystemps}).
%
\input{GerlichTscheuschner2007a_370_co2_greenhouses_radiative_table012_}

This is no accident but a consequence of
H\"older's inequality\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Hoelder1889,Hardy1934,Beckenbach1961,Kuptsov2001}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{equation}
\int_X fg\,d\mu
\le
\left\{
\int_X f^p\,d\mu
\right\}
^{1/p}
\cdot
\left\{
\int_X g^q\,d\mu
\right\}
^{1/q}
\end{equation}
%
for two non-negative measurable functions
$f$, $g$ and non-negative integers $p$, $q$
obeying
%
\begin{equation}
\frac{1}{p}
+
\frac{1}{q}
=
1
\end{equation}
%
In the case discussed here
one has
%
\begin{equation}
p=4,
\phantom{x} 
q=4/3,
\phantom{x}
g(x) \equiv 1
\end{equation}
%
and 
%
\begin{equation}
f = T
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsubsection{Non-existence of the natural greenhouse effect}
%
According to the consensus among global climatologists
one takes the $-18^\circ{\rm C}$ computed from
the $T^4$ average and compares it to the
fictitious Earth's average temperature of $+15\,^\circ{\rm C}$.
The difference of  $33\,^\circ{\rm C}$ is attributed to the
\textit{natural greenhouse effect}. 
As seen in Equation (\ref{Eq:Tphys}) a correct averaging yields a temperature
of $-129\,^\circ{\rm C}$. Evidently, something must be 
fundamentally wrong here.

In global climatology temperatures are computed from given
radiation intensities, and this exchanges cause and effect. 
The current \textit{local} temperatures determine the radiation
intensities and not  \textit{vice versa}. If the soil is warmed up by
the solar radiation many different local processes are triggered,
which depend on the local movement of the air, rain,
evaporation, moistness, and on the local ground conditions
as water, ice, rock, sand, forests, meadows, etc.\
One square meter of a meadow does not know anything of
the rest of the Earth's surface, which determine the
global mean value. 
Thus, the radiation is \textit{locally} 
determined by the \textit{local}
temperature.
Neither is there a global radiation balance, 
nor a global radiation budget, even in the mean-field limit.

While it is incorrect to determine a temperature from 
a given radiation intensity, one is allowed to compute 
an effective radiation temperature
%
$T_{\mbox{\scriptsize\rm eff rad}}$ 
%
from $T^4$ averages
representing a mean radiation emitted from the Earth
and to compare it with an assumed Earth's average 
temperature
%
$T_{\mbox{\scriptsize\rm mean}}$
%
H\"older's inequality says that the 
former is always larger than the latter
%
\begin{equation}
T_{\mbox{\scriptsize\rm eff rad}} 
> 
T_{\mbox{\scriptsize\rm mean}}
\end{equation}
%
provided sample selection 
and averaging (probability space)
remain the same. 

For example,
if $n$ weather stations distributed around 
the globe measure $n$ temperature values
$T_1$, \dots $T_n$, an \textit{empirical} 
mean temperature will be defined as
%
\begin{equation}
T_{\mbox{\scriptsize\rm mean}}
=
\frac{1}{n}
\sum_{i=1}^{n}
T_i
\end{equation}
% 
For the corresponding black body radiation intensity 
one can approximately set
%
\begin{equation}
S_{\mbox{\scriptsize\rm mean}}
=
\frac{1}{n}
\sum_{i=1}^{n}
\sigma \, T_i^4
=:
\sigma \, 
T_{\mbox{\scriptsize\rm eff rad}}^4
\end{equation}
%
defining an \textit{effective} radiation temperature
%
\begin{equation}
T_{\mbox{\scriptsize\rm eff rad}}
=
\sqrt{
\frac{1}{\sigma} 
S_{\mbox{\scriptsize\rm mean}}
} 
\end{equation}
%
One gets immediately
%
\begin{equation}
T_{\mbox{\scriptsize\rm eff rad}} 
=
\sqrt[4]{
\frac{1}{n}
\sum_{i=1}^{n}
T_i^4
}
\label{Eq:SqrtSumT4}
\end{equation} 
%
H\"older's inequality shows that one always has 
%
\begin{equation}
T_{\mbox{\scriptsize\rm eff rad}}
>
T_{\mbox{\scriptsize\rm mean}}  
\end{equation}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsubsection{A numerical example}
%
From 
Equation~(\ref{Eq:SqrtSumT4}) 
one can construct numerical examples 
where e.g.\ a few high local temperatures 
spoil an average built from a large 
collection of low temperatures.
A more realistic distribution 
is listed in Table~\ref{Table:numerical}.
The effective radiation temperature
$T_{\mbox{\scriptsize\rm eff rad}}$ 
is slightly higher than the 
average
$T_{\mbox{\scriptsize\rm mean}}$
of the measured temperatures.
According to H\"older's inequality 
this will always be the case.
%
\input{GerlichTscheuschner2007a_370_co2_greenhouses_radiative_table013_.tex}
%  
  
Thus there is no longer any room 
for a \textit{natural greenhouse effect}, 
both mathematically and physically:
%
\begin{itemize}
\item Departing from the 
      \textit{physically incorrect} 
      assumption of radiative balance a
      \textit{mathematically correct}
      calculation of the average temperature
      lets the difference temperature
      that defines the natural greenhouse effect
      explode.
\item Departing from the 
      \textit{mathematically correct}
      averages of 
      \textit{physically correct} 
      temperatures (i.e.\ measured temperatures)
      the corresponding effective radiation temperature
      will be \textit{always higher} than the  
      average of the measured temperatures. 
\end{itemize}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
\subsubsection{Non-existence of a global temperature}
%
In the preceding sections mathematical and physical arguments have been presented
that the notion of a global  temperature is meaningless.
Recently, 
Essex, McKitrick, and Andresen showed 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Essex2007}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{quote}
\lq\lq
that there is no physically meaningful 
global temperature for the Earth in the 
context of the issue of global warming. 
While it is always possible to construct 
statistics for any given set of
local temperature data, an infinite range 
of such statistics is mathematically permissible
if physical principles provide no explicit basis 
for choosing among them. Distinct and
equally valid statistical rules can and 
do show opposite trends when applied to the
results of computations from physical models 
and real data in the atmosphere. A given
temperature field can be interpreted 
as both \lq warming\rq\ and \lq cooling\rq\ 
simultaneously, making the concept of warming 
in the context of the issue of global warming 
physically ill-posed.\rq\rq
\end{quote}
%
Regardless of any ambiguities, a global mean temperature 
could only emerge out of many local temperatures. Without knowledge
of any science everybody can see, how such a changing average 
near-ground temperature is constructed: 
There is more or less sunshine on the ground 
due to the distribution of clouds.
This determines a field of local 
near-ground temperatures, 
which in turn determines the change
of the distribution of clouds and, hence, 
the change of the temperature average, which 
is evidently independent of the carbon dioxide 
concentration.
Mathematically, an evolution of a temperature distribution 
may be phenomenologically described by a differential
equation. The averages are computed  afterwards 
from the solution of this equation. However, one
cannot write down a differential equation directly 
for averages.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
\subsubsection{The rotating globe}
%
Since the time when Fourier formulated
the heat conduction equation,
a non-linear boundary condition
describing radiative transfer
of a globe with a sun-side and
a dark side has never belonged 
to the family of solvable heat conduction problems, 
even in the case of a non-rotating globe.

Regardless of solvability, one can
write down the corresponding equations
as well as their boundary conditions.  
%
\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_Earthball_rotating_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_Earthball_rotating_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_Earthball_rotating_.png}}
}{}
\vspace*{8pt}
\caption{The rotating globe}
\label{fig:rotatingglobe}
\end{figure}
%
If a rotating globe 
(Fig.~\ref{fig:rotatingglobe}) 
was exposed to radiation and only 
radiative heat transfer to its environment
was possible, the initial problem 
of the heat conduction equation 
would have to be solved with the 
following boundary condition
%
\begin{equation}
- \lambda\,
\frac{\partial T}{\partial\textbf{n}}
=
\left\{\begin{array}{lcc}
\sigma T^4 - {\rm S} \cdot \sin\vartheta \cos(\varphi-\omega_d t) 
&
\mbox{{\rm \ \ \ if\ }}
&
- \pi/2 \leq \varphi - \omega_d t \leq \pi/2
\\
\sigma T^4
& 
\mbox{{\rm \ \ \ if\ }}
&
\phantom{xx}
\pi/2 \leq \varphi - \omega_d t \leq 3\pi/2
\end{array}\right.
\end{equation}
%
where
%
\begin{equation}
\frac{\partial}{\partial\textbf{n}}
=
\textbf{n} \cdot \mbox{\boldmath$\nabla$}
\end{equation}
%
denotes the usual normal derivative
at the surface of the sphere
and $\omega_d$ the angular frequency 
associated with the day-night cycle.
By defining an appropriate geometry factor
%
\begin{equation}
\zeta(\vartheta,\varphi, \omega_d, t) = \sin\vartheta \cos(\varphi-\omega_d t) 
\end{equation}
%
and the corresponding Sun side area 
%
\begin{equation}
A 
= 
\lbrace 
(\varphi,\vartheta) 
\,|\,  
\zeta(\vartheta,\varphi,\omega_d, t)
\geq 0 
\rbrace
\end{equation} 
%
one can rewrite the expression as  
%
\begin{equation}
- \lambda\,
\frac{\partial T}{\partial\textbf{n}}
=
\left\{\begin{array}{lcc}
\sigma T^4 - {\rm S} \cdot \zeta(\vartheta,\varphi, \omega_d, t) 
&
\mbox{{\rm \ \ \ if\ }}
&
(\varphi,\vartheta) \in A
\\
\sigma T^4
& 
\mbox{{\rm \ \ \ if\ }}
&
(\varphi,\vartheta) \not\in A
\end{array}\right.
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsubsection{The obliquely rotating globe}
%
The result obtained above may be generalized 
to the case of an obliquely rotating globe. 
%
\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_Earthball_rotating_oblique_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_Earthball_rotating_oblique_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_Earthball_rotating_oblique_.png}}
}{}
\vspace*{8pt}
\caption{An obliquely rotating globe}
\label{fig:obliquelyrotatingglobe}
\end{figure}
%

\noindent%
For an obliquely rotating globe 
(Fig.~\ref{fig:obliquelyrotatingglobe})
one has
%
\begin{equation}
- \lambda\,
\frac{\partial T}{\partial\textbf{n}}
=
\left\{\begin{array}{lcc}
\sigma T^4 - {\rm S} \cdot \xi(\vartheta_0,\vartheta,\varphi,\omega_y,\omega_d,t) 
&
\mbox{{\rm \ \ \ if\ }}
&
(\varphi,\vartheta) \in A
\\
\sigma T^4
& 
\mbox{{\rm \ \ \ if\ }}
&
(\varphi,\vartheta) \not\in A
\end{array}\right.
\end{equation}
%
where $\partial/\partial\textbf{n}$ 
denotes the usual normal derivative
on the surface of the sphere 
and $\omega_y$, $\omega_d$ the angular frequencies
with the year cycle and the day-night cycle,
respectively.%
%
\footnote{Here sidereal time is used\TschSpace%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{%
DictionaryGeophysicsAstronomyAstrophysics2001,%
EncyclopaediaAstronomyAndAstrophysics2001%
}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
} 
%
The geometry factor now reads
%
\begin{eqnarray}
\xi(\vartheta_0,\vartheta,\varphi,\omega_y,\omega_d,t)
&=&  
\,\,\,\,\,\,
[
\,\,\,\,\, 
\sin(\omega_y t) \cos(\omega_d t) + \cos(\omega_y t)\sin(\omega_d t)\cos \vartheta_0 
]
\, 
\sin\vartheta\cos\varphi 
\nonumber \\
&&
+
\,
[ 
-
\sin(\omega_y t) \sin(\omega_d t) + \cos(\omega_y t)\cos(\omega_d t)\cos \vartheta_0 
]
\, 
\sin\vartheta\sin\varphi
\nonumber \\
&&
-
\,
[
\,
\cos(\omega_y t)\sin\vartheta_0
\,
]
\,
\cos\vartheta
\end{eqnarray}
%
and the expression for the sun-side surface
is given by
%
\begin{equation}
A 
= 
\lbrace 
(\varphi,\vartheta) 
\,|\,  
\xi(\vartheta_0,\vartheta,\varphi,\omega_y,\omega_d,t)
\geq 0
\rbrace 
\end{equation} 
%
Already the first unrealistic problem 
will be too much for any computer.
The latter more realistic model 
cannot be tackled at all.
The reasons for this is not only 
the extremely different frequencies
$\omega_y$ and $\omega_d$ but also
a very non-physical feature 
which affects the numeric as well: 
According to a famous law formulated by
Wiener, 
almost all particles in this 
mathematical model which cause 
the diffusion, move on paths 
at infinitely high speeds\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Bauer1964,Bauer2002}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Rough estimates indicate that even these
oversimplified problems cannot be tackled
with any computer.
Taking a sphere with dimensions of the Earth
it will be impossible to solve this problem 
numerically even in the far future.
Not only the computer would work ages,
before a \lq\lq balanced\rq\rq\ temperature 
distribution would be reached,
but also the correct initial temperature
distributions could not be determined
at all.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
\subsubsection{The radiating bulk}
%
The physical situation of a radiating 
volume where the radiation density
%
\begin{equation}
{\rm S}(T) = \sigma T^4
\end{equation}
%
emitted through the surface shell 
originates from the volume's heat content, 
cannot be realized easily, if at all.
However, it is interesting to study 
such a toy model in order to get a feeling about 
radiative equilibration processes which
are assumed to take place within a
reasonable time interval.
  
With disregard to the balancing processes 
inside, one gets the differential equation
%
\begin{equation}
{\rm V}  
\varrho\,\,
c_{\rm v}
\frac{dT}{dt} 
=
- \Omega \, \sigma T^4
\end{equation}
%
with 
${\rm V}$ denoting the volume,
$\varrho$ the density,
$c_{\rm v}$ the isochoric specific heat,
$\Omega$ the surface of the body.
By defining 
%
\begin{equation}
\eta = \frac{\Omega}{{\rm V}}
\end{equation}
%
the above equation can be rewritten as
%
\begin{equation}
\frac{dT}{dt} 
=
-
\frac{\eta\,\sigma}{\varrho\,c_{\rm v}}
\cdot T^4
\end{equation}
%
For a cube with an edge length of $a$
one has $\eta = 6/a$,
for a globe with radius $r$
one has $\eta = 3/r$ instead.
For bodies with unit volumes
$\eta = 6$ or $\eta = 4.8$,
respectively. 

The differential equation is easily solvable.
The solution reads
%
\begin{equation}
T(t) 
=
T_0
/
{
\sqrt[3]
{
1
+
\frac{3\,\eta\,\sigma T_0^3}
     {\varrho\,c_{\rm v}}
\, t
}
}
\end{equation}
%
At an initial temperature of 
$300\,{\rm K}$ 
with the values of 
$\varrho$ and $c_{\rm v}$ 
for air, one gets one half of the temperature value 
within three seconds for the standard cube
(cf.\ Figure~\ref{Fig:CubeCooling})
%
\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_cooling_.eps,width=3.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_cooling_.png}}
\vspace*{-8pt}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_cooling_.png}}
\vspace*{-8pt}
}{}
\vspace*{8pt}
\caption{The cooling curve for a 
         radiating standard cube}
\label{Fig:CubeCooling}         
\end{figure}
%
For iron the 
isochoric thermal diffusivity 
%
\begin{equation}
a_{\rm v}=\varrho\,c_{\rm v}
\end{equation}
% 
is about 3000 times 
higher than for air, 
the half time for the temperature 
decrease is approximately three hours.
For air, even if only one of the cube's planes 
were allowed to radiate, one would get 
a fall in temperatures of seventy degrees 
within the first three seconds, 
and almost 290 degrees within ten hours 
- a totally unrealistic cooling processes.
 
Hence, this simple assessment will prove
that one has to be extremely careful, 
if the radiation laws for black-body 
radiation, where the energy comes 
from the heated walls of the cavity, 
are to be used for gases, where the 
emitted electromagnetic radiation 
should originate from the movements 
of the gas molecules 
(cf.\ Section~\ref{Sec:Radiation}).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
\subsubsection{The comprehensive work of Schack}
%
Professor Alfred Schack, 
the author of a standard textbook 
on industrial heat transfer\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{AlfredSchackBook}, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
was the first scientist who pointed out 
in the twenties of the past century
that the infrared light absorbing fire gas components 
\textit{carbon dioxide} (${\rm CO}_2$) 
and 
\textit{water vapor} (${\rm H}_2{\rm O}$) 
may be responsible 
for a higher heat transfer in the combustion 
chamber \textit{at high burning temperatures} 
through \textit{an increased emission in the infrared}.
He estimated the emissions by measuring 
the spectral absorption capacity of 
carbon dioxide and water vapor.
  
In the year 1972 Schack published a paper in
Physikalische Bl\"atter entitled 
\lq\lq The influence of the carbon dioxide content
of the air on the world's climate\rq\rq.
With his article he got involved in the 
climate discussion and emphasized
the important role of water vapor\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{AlfredSchack1972}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 

Firstly, Schack estimated the mass 
of the consumed fossil fuels up
%
\begin{equation}
m_{ 
\mbox{\scriptsize\rm burned}
}
=
5 \cdot 10^{12}\,{\rm kg}
=
5 \, {\rm GtC}
\end{equation}
%
\textit{per anno}. 
Since $1\,{\rm kg}$ produces 
$10\,{\rm m}^3$ waste gas 
with $15\,\%$ ${\rm CO}_2$,
a volume of 
%
\begin{equation}
V_{
\mbox{\scriptsize{${\rm CO}_2$}}
}
= 
7.5 \cdot 10^{12}\,{\rm m}^3 
\end{equation}
%
is blown into the Earth's atmosphere,
whose total volume under normal conditions 
($0\,^\circ{\rm C}$ and 760\,{\rm mm}\,{\rm Hg})
is  
%
\begin{equation}
V_{ 
\mbox{\scriptsize\rm atmosphere} 
} 
= 
4 \cdot 10^{18}\,{\rm m}^3 
\end{equation}
%
It follows immediately that the increase 
of the ${\rm CO}_2$ concentration
is approximately 
$1.9\cdot 10^{-6}$
\textit{per anno}. 
About one half is absorbed by the oceans, 
such that the increase of ${\rm CO}_2$
is reduced to
%
\begin{equation}
\frac
{
\Delta
V_{
\mbox{\scriptsize{${\rm CO}_2$}}
}
}
{
V_{
\mbox{\scriptsize{${\rm CO}_2$}}
}
}
=
0.95\cdot 10^{-6}
\end{equation}
%
\textit{per anno.}

With the \lq\lq current\rq\rq\ (1972) atmospheric 
${\rm CO}_2$ 
volume concentration of 
%
\begin{equation}
0.03\,\% = 300 \cdot 10^{-6}
\end{equation}
% 
and an relative annual increase of
%
\begin{equation}
0.32\,\% 
=
\frac
{0.95 \cdot 10^{-6}}
{300 \cdot 10^{-6}}
\end{equation}
% 
the ${\rm CO}_2$ concentration in the atmosphere 
would rise by one third of current
concentration within 100 years, 
supposed the fossil fuel consumption
will remain constant.

Schack then shows that ${\rm CO}_2$ 
would absorb only one seventh of the ground's 
heat radiation at most, if the water vapor had 
not already absorbed the infrared light 
in most situations. Furthermore, 
a doubling of the ${\rm CO}_2$-content 
in the air would only halve the 
radiation's characteristic absorption length, 
that is, the radiation would be absorbed 
at a length of $5\,{\rm km}$ instead of 
at a length of $10\,{\rm km}$, for example.

Schack discussed the ${\rm CO}_2$ contribution
only under the aspect that ${\rm CO}_2$ acts
as an absorbent medium.
He did not get the absurd idea 
to heat the radiating warmer ground 
with the radiation absorbed and re-radiated
by the gas. 

In a comment on an article by 
the science journalist Rudzinski\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Rudzinski1976}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
the climatologist Oeschger 
objectioned against Schack's analysis 
of the influence of the 
${\rm CO}_2$ concentration 
on the climate that Schack had not calculated 
thoroughly  enough\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Oeschger1976}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
In particular, he referred to radiation transport calculations.
However, such calculations have formerly been performed  
only for the atmospheres of stars, because the processes 
in planetary atmospheres are far too complicated 
for such simple models. The goal of astrophysical 
radiation transport calculations is to calculate as many 
absorption lines as possible with one boundary density 
distribution and one temperature dependency with respect 
to the height with Saha's equation
and many other additional hypotheses\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Unsoeld1955}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
However, the boundary density of the radiation intensity 
cannot be derived from these calculations. 

One should emphasize that Schack was the first 
scientist to take into account the selective 
emission by the infrared light absorbing 
fire-gases for combustion chambers. 
Therefore one is driven to the verge of irritation 
when global climatologists blame 
him for not calculating complicatedly enough, 
simply because he saw the primitive 
physical concepts behind the equations for the 
radiation transfer. 
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
